The importance of the future

It would be bad for me to permanently cease to exist in five minutes. But why? Suppose first a metaphysics of time on which there is no future, namely Growing Block or Presentism. On such a metaphysics there is no such thing as my future life, so how could it be bad for there to be a cessation of it?

Since the only tenable alternative to Growing Block and Presentism is Eternalism, the view that the past and future are real (oddly, there are no Futurists who think the future is real but deny the reality of the past), Eternalism is true.

Now, given Eternalism, we have a choice for three visions of our persistence through time. On one vision, Exdurantism, we are instantaneous stages that do not persist through time at all—at most we have temporal counterparts at other times. This does not fit with the intuition of my radical incompleteness should I cease to exist in five minutes. The second vision is Endurantism: I am wholly present at each time at which I exist. But then if the present moment is real, and eternally will be real, and I wholly exist at this present moment, then the intuition about the deep incompleteness I would have were my existence to permanently end in five minutes is undercut. So that can't be right either.

What remains is a family of views on which we are strung out four-dimensionally. The most common member of the family is Perdurantism: I am four-dimensional but have three-dimensional stages localized at times. A less common view is that I am four-dimensional, but not divided up into stages. Both of these views do justice to the idea that my existence is deeply incomplete, in something like the way it would be if I were missing an arm, should I cease to exist in five minutes.

As far back as I thought much about time (probably going back to age 10) I was an Eternalist. Until a couple of years ago, I was an Endurantist. Then I started being unsure whether Endurantism or a stageless four-dimensional view is right. The above argument strongly pushes me towards a four-dimensional view, and since I don't believe in stages, a stageless one.

Moreover, the above may help with a puzzle I used to have, which was how a B-Theorist should think about the badness of impending evils (especially death). How can a B-Theorist make sense of the badness of being closer and closer to something bad? But that may primarily be a problem for the Endurantist, since the Endurantist thinks we are three-dimensional beings wholly located in the here and now (as well as in the there and later, of course).

Hope

If there are ten lottery tickets, and I hold one, I shouldn't hope to win, but I should simply assign probability 1/10 to my winning. Anything beyond the probabilities in the way of hope would be irrational. Likewise, if I have probability 9/10 of winning. Then I can have confidence, but this confidence should no more be a hope than in the former case. It's just a confidence of 9/10.

But if my friend has fallen morally many times but promises to do better, I shouldn't simply calculate the probability of his doing better using the best inductive logic and leave it at that. I should hope he will do better.

What makes for the difference? In the case of the friend, he should do better. But it is, of course, false that I should win the lottery. Indeed, the outcome of my winning the lottery is in no way normatively picked out. I can appropriately hope that the lottery will be run fairly, but that's that.

If this is right then it seems hope is of what should be. Well, that's not quite right. For if I have done something so terrible that my friend is under no obligation to forgive me, I can still hope for her supererogatory forgiveness. So, perhaps, hope is of what should be or what goes over and beyond a should.

If this is right, then this neatly dovetails with my account of trust or faith. Faith has as its proper object a present state of affairs that should be, such as a testifier's honesty and reliability, or perhaps—I now add—a present state of affairs that goes beyond a should. Hope has as its proper object a future state of affairs that should be or goes beyond a should. Both of these flow from love.

If this is right, then in order for there to be appropriate hope in things beyond human power—such as a hope that an asteroid won't wipe out all life on earth—there must be shoulds, or beyond-shoulds, that go beyond human power. This requires an Aristotelian teleology or theism.

Despair

1. If there is no hope of an afterlife, all is hopeless.
2. If all is hopeless, ultimate despair is the right attitude.
3. Ultimate despair is not the right attitude.
4. So there is hope of an afterlife.

I will argue for 1 and 3. If there is no hope of an afterlife, any redeeming value we might hope for is overshadowed by the ultimate evil of both the end of our individual lives and of the human race. But despair is not the right attitude, since despair makes it impossible for us to live our moral lives, both in terms of the motivation to pursue the good and in our duty to comfort others. Despair saps our motivation. And faced with ultimate hopelessness, any comfort we might offer to others is insincere and dishonest. In despair at ultimate hopelessness, we could only live the good human life through self-deceit and the deceit of others. But that is not right. So ultimate despair cannot be the right attitude as it makes the good life impossible.

So there is hope.

Deep Thoughts XXXV

Meeting the minimum requirements is always good enough.

[This is a variant on XXXIV. There are times when we say things like "The minimum is not good enough." When we do that, what I think happens is that we have a context shift. "The minimum" is understood relative to one set of ends or norms while the "good enough" is understood relative to another. One kind of a case is where you're competing for a job. Meeting the minimum required qualifications is good enough for being hirable in principle (if it's not, the minimum requirements were incorrectly stated), but is not good enough for beating the competition. Another kind of case is where someone is being evaluated in a number of areas (or with respect to a multiplicity of assignments). In each area, there is a minimum requirement. Meeting that requirement is good enough for not failing according to that requirement. But there may be a second, meta requirement to exceed the minimum in most of the areas. (There cannot coherently be a requirement to exceed the minimum in all areas. For if there were such, then the "minimum" in each area would not be a minimum requirement but a maximum disqualifier.) In any case, when we keep the context constant (and as a rule in natural language context in short sentences stays constant), "The minimum is not good enough" is a self-contradiction.]

A simple consequence argument

Say that p and q are nomically equivalent provided that the laws of nature entail that p holds if and only if q does.

Assume:

1. If q is not up to you, and p is nomically equivalent to q, then p is not up to you.

Suppose determinism. Let L be the laws. Let t₀ be 1000 years ago. Let p be a proposition reporting something you do. Let q be the disjunction of all the nomically possible states of the universe at t₀ that evolve under L in such a way as to make p true. Then, plausibly:

2. p and q are nomically equivalent.

For given the deterministic laws, if p is true, then a thousand years ago the universe must have been such as to have to evolve to make p true.[note 1] And conversely, the laws entail that if it was such, then p is true.

Finally, observe that events a thousand years ago aren't up to you:

3. q is not up to you.

We conclude that p is not up to you. So no actions are up to you if determinism holds.

Substance causation, agent causation and time

Aristotelians about causation think all causation is substance causation. Events are causes only derivatively. What does the real causing are substances. This should make Aristotelians very sympathetic to the use of agent causation in the theory of free will. And insofar as the theory of agent causation is just that the agent is the cause of free actions, the Aristotelian who believes that we are substances[note 1] is surely going to agree that we are the causes of our free actions, and we are both agents and substances, so the agent is the cause of her free actions.

So far so good. But there is more to agent causation in regard to free will. Typically, agent causalists invoke agent causation to solve problems such as the randomness problem for libertarians. Agent causation is what makes an action be genuinely one's own action rather than a random blip. But the Aristotelian's embrace of substance causation is too broad. For not only does the Aristotelian think that her free actions are caused by her, she also thinks her non-free actions are caused by her, and even things like the circulation of the blood, which isn't an action at all, are caused by her. Moreover, since she is an agent, she thus thinks all of these things are caused by an agent. But if agent causation metaphysically lumps free actions with non-free ones, and doesn't distinguish them metaphysically from the circulation of the blood, then agent causation can't do the job it's designed for. The Aristotelian believes in agent causation, of course, and may do so with good metaphysical reason, but this agent causation cannot be used to solve the problems that the free will theorists want it to solve.

This line of thought might lead some Aristotelians about causation to accept a version of Cartesian dualism on which we are souls. For then one might hold, contrary to Aristotle, that only our free actions are caused by us and that the circulation of the blood and so on is not caused by us, because we are immaterial beings whose only direct effects are in whatever the equivalent of the pineal gland on this theory will be. This is not in the Aristotelian spirit, though, and it leads to unhappy ethical conclusions (bodies as akin to property).

I think there is something else one should say here. One shouldn't say that agent causation just is causation by an agent. Rather, agent causation is causation by an agent qua agent. You cause your free actions qua agent and you circulate your blood qua mammal, though of course you are both agent and mammal. It is a bit odd to say that you don't perform your non-free actions qua agent, though. After all, how can there be an action without an agent? Aren't all actions, free or not, the actions of an agent qua agent? Maybe. But maybe the distinction is still of some help, for maybe the kinds of mere randomness we want to rule out with the distinction isn't an action at all when looked at more closely.

There is another issue around here. There needs to be more to substance causation than the simple structure substance x causes event E. For paradigmatic substances persist over a long time, but many of their effects happen only at particular times in their existence. And there is an explanation of why the substance causes an effect at one time or another. For instance, I caused oatmeal to be assimilated to me earlier to day because I was hungry. The explanation includes not just the substance, but a state of a substance (and maybe other substances—but that's for another day). I caused the assimilation of my breakfast not only qua agent, but qua hungry agent. Likewise, I circulate the blood not just qua mammal, but qua mammal with a brain stem that sends such-and-such electrical signals to the heart.

Apart from considerations of free will, then, Aristotelians should say that the structure of substance causation is something like: substance x qua in state S causes event E. If we have an Aristotelian constituent ontology, then the state will be a mode (an essence, a necessary accident or a contingent accident) of the substance, and the causation relation will be a ternary relation between the substance, the mode and the event.

But now that we have all of this detail in place, we can go back and ask whether we still get benefits of an agent causal theory in regard to free will. That's not so clear. For instance, when I qua hungry agent caused my breakfast to be assimilated to me, the work distinguishing this from non-actions like circulation is being done by my state of hungry agency. It is because this state is involved in the causation—and not just involved, but involved in the right way (my being a hungry agent could cause me to grow a tail if I was rigged the wrong way)—that I am acting. But event causalists can say something exactly parallel. What distinguishes my eating the breakfast from my circulating the blood is that the former is caused by the event of my being a hungry agent qua my being a hungry agent.

So I do not know that Aristotelian agent causalists can claim to do better than event causalists. In fact, for certain ends they might well want to join cause with the event causalists.

Pascal's Wager in a social context

One of our graduate students, Matt Wilson, suggested an analogy between Pascal's Wager and the question about whether to promote or fight theistic beliefs in a social context (and he let me cite this here).

This made me think. (I don't know what of the following would be endorsed by Wilson.) The main objections to Pascal's Wager are:

1. Difficulties in dealing with infinite utilities. That's merely technical (I say).
2. Many gods.
3. Practical difficulties in convincing oneself to sincerely believe what one has no evidence for.
4. The lack of epistemic integrity in believing without evidence.
5. Would God reward someone who believes on such mercenary grounds?
6. The argument just seems too mercenary!

Do these hold in the social context, where I am trying to decide whether to promote theism among others? If theistic belief non-infinitesimally increases the chance of other people getting infinite benefits, without any corresponding increase in the probability of infinite harms, then that should yield very good moral reason to promote theistic belief. Indeed, given utilitarianism, it seems to yield a duty to promote theism.

But suppose that instead of asking what I should do to get myself to believe the question is what I should try to get others to believe. Then there are straightforward answers to the analogue of (3): I can offer arguments for and refute arguments against theism, and help promote a culture in which theistic belief is normative. How far I can do this is, of course, dependent on my particular skills and social position, but most of us can do at least a little, either to help others to come to believe or at least to maintain their belief.

Moreover, objection (4) works differently. For the Wager now isn't an argument for believing theism, but an argument for increasing the number of people who believe. Still, there is force to an analogue to (4). It seems that there is a lack of integrity in promoting a belief that one does not hold. One is withholding evidence from others and presenting what one takes to be a slanted position (for if one thought that the balance of the evidence favored theism, then one wouldn't need any such Wager). So (4) has significant force, maybe even more force than in the individual case. Though of course if utilitarianism is true, that force disappears.

Objections (5) and (6) disappear completely, though. For there need be nothing mercenary about the believers any more, and the promoter of theistic beliefs is being unselfish rather than mercenary. The social Pascal's Wager is very much a morally-based argument.

Objections (1) and (2) may not be changed very much. Though note that in the social context there is a hedging-of-the-bets strategy available for (2). Instead of promoting a particular brand of theism, one might instead fight atheism, leaving it to others to figure out which kind of theist they want to be. Hopefully at least some theists get right the brand of theism—while surely no atheist does.

I think the integrity objection is the most serious one. But that one largely disappears when instead of considering the argument for promoting theism, one considers the argument against promoting atheism. For while it could well be a lack of moral integrity to promote one-sided arguments, there is no lack of integrity in refraining from promoting one's beliefs when one thinks the promotion of these beliefs is too risky. For instance, suppose I am 99.99% sure that my new nuclear reactor design is safe. But 99.9999% is just not good enough for a nuclear reactor design! I therefore might choose not promote my belief about the safety of the design, even with the 99.9999% qualifier, because politicians and reporters who aren't good in reasoning about expected utilities might erroneously conclude not just that it's probably safe (which it probably is), but that it should be implemented. And the harms of that would be too great. Prudence might well require me to be silent about evidence in cases where the risks are asymmetrical, as in the nuclear reactor case where the harm of people coming to believe that it's safe when it's unsafe so greatly outweighs the harm of people coming to believe that it's unsafe when it's safe. But the case of theism is quite parallel.

Thus, consistent utilitarian atheists will promote theism. (Yes, I think that's a reductio of utilitarianism!) But even apart from utilitarianism, no atheist should promote atheism.

Conditional probability and probability comparisons

Suppose we want to say that event B is more likely than another A. What does that mean? A natural thing to say is that P(B)>P(A). But that doesn't fit our intuitions. For all measure zero sets then end up being equally likely. We can get a sharper comparison if we instead of starting with unconditional probabilities, we work with primitive conditional probabilities. A natural way that I've considered in the past is to say that A≤B if and only if P(A|A∪B)≤P(B|A∪B). This lets you compare tiny sets, like a single point to two points. But this approach has the intuitive disadvantage that then if we use uniform measure on [0,1], then [0,1]≤[0,1).

But there is a better way to generate a comparison from a conditional probability. Say that A≤B if and only if the two sets are identical or P(A−B|(A−B)∪(B−A))≤P(B−A|(A−B)∪(B−A)). It's not that hard (unless one is as sleepy as I am this morning) to show that this relation is reflexive, transitive and total—i.e., a weak order. Moreover, this weak order has the property that if A is a proper subset of B, then we're guaranteed to have strict inequality: A<B.

Update: A. Paul Pedersen informs me that De Finetti gives a definition equivalent to this on page 367 of the second volume of his probability book.

There is no regular approximately invariant finitely additive probability measure on all subsets of a cube or ball

For a totally ordered field K, say a hyperreal one, write x≈y (and say that they are approximately equal) provided that x−y is 0 or infinitesimal. A K-valued probability P defined for all subsets of Ω is said to be regular provided that P(A)>0 whenever A is non-empty. It is approximately rigid motion invariant provided that P(A)≈P(gA) for every rigid motion g and set A such that A∪gA⊆Ω. The following can be proved in Zermelo-Fraenkel (ZF) set theory without any Axiom of Choice:

Theorem 1. There is no totally ordered field K and a regular K-valued approximately rigid motion invariant finitely additive probability on all subsets of a ball or cube Ω.

If we delete "approximately", this follows from this.

The result follows from this post. Given such a regular measure we can define a preorder ≤ by letting A≤B if and only if P(A)≤P(B). By the Theorem from that post, it follows in ZF that Banach-Tarski is true. But Banach-Tarski implies that there is no approximately rigid motion invariant finitely additive probability on all subsets of a ball or cube.

(Why ball or cube? This saves me from having to worry about some edge effects given our definition of invariance.)

Another result, proved by similar methods:

Theorem 2. Let Ω be a subset of three-dimensional Euclidean space invariant under rotations about the origin 0. If K is a totally ordered field and P is a regular K-valued finitely additive probability on all subsets of Ω approximately invariant under rotations about the origin, then P({0})≈1.

Suppose now that we have a particle undergoing Brownian motion released at time t₀ at the origin, and then observed at time t₁. The probability of its being in some set at t₁ should be at least approximately invariant under rotations, and of course it is unacceptable to say that the probability that it is at the origin is approximately one—on the contrary, with approximately unit probability it is going to be away from the origin.

Update: Similar things hold for full-conditional probabilities, where approximate invariance is replaced with
invariance conditionally on the whole space (but there is no requirement of invariance conditionally on subsets of the space).

Interesting riddle concerning the Axiom of Choice

There is an interesting riddle over at MathOverflow concerning the Axiom of Choice.

Love of God is needed, not just love of neighbor

Suppose Jim believes that he has a long-lost elder brother, and in order to become the next Marquess of Winchester he strives to hunt down and murder that brother. The above is incompatible with Jim's being virtuous, but it is logically compatible with (though psychologically unlikely to coexist with) Jim's loving every human being, since Jim's belief might be false, and he might thus have no long-lost brother, and hence no failure of love for that brother. Thus, loving each human being does not entail being virtuous.

But loving God—that's a different matter. For if one loves God, then one is thereby disposed to love all that God has made. Thus, Jim, while he does not fail in love of any particular neighbor, does fail in love for God.

Theologians who say "God doesn't exist"

Some theologians say that God doesn't exist--God is beyond being. Here is one way to make sense of their claim: Non-relational claims with "God" as the subject term have the word "God" functioning like the "It" in "It's raining" (we should think of "It's raining" on this reading as a nullary predicate). In other words, what we are really expressing are subjectless claims. When the vulgar believer says "God doesn't exist", that's not literally true. What is literally true is something like: "It's Godding". And when the vulgar believer says "God is wise", what's literally true is: "It's Godding wisely" just as "The rain is intense" really means that it's raining intensely. Relational claims can be similarly handled. "The rain falls on me" is more precisely expressed as "It's raining on me", and "God creates the earth" is better said as "It's Godding creatively with respect to the earth." A theologian of this stripe can then talk with the vulgar, but she has a preferred explication of what is being said.

I think this story fails when we talk about love between God and humans. For love is essentially relational. An account of love that eliminates either the subject term or the object term is automatically not an account of love in the proper sense. (There is an extended sense in which someone might be said love a non-existent person. But I think it's more proper to say that she seems to love. If presentism and no-afterlife are true, then in this example, Sally does not love Fred—she only thinks she does.)

Loving without knowing

Fred is lost in the desert and dies. Without knowing this, Sally, his loving wife who is a presentist and thinks there is no afterlife spends weeks searching for Fred in the desert, in uncertainty whether Fred is still alive, despite great hardship and danger to her own life.

Observe that a presentist who disbelieves in an afterlife thinks that the dead are simply nonexistent. So Sally is not only uncertain whether Fred is alive, but she is uncertain whether Fred exists. Yet she acts out of love. Hence:

• It is possible to love someone while being unsure whether he exists.

(Of course, one might also think that this case points to there being something defective in believing in presentism or in doubting an afterlife. )

Better than perfect

A necessary and sufficient condition for a student to have perfect performance on a calculus exam is to correctly, perfectly clearly and with perfect elegance answer every question within the time allotted. But what if the student also includes her proof of the Riemann Zeta Conjecture on the last page? Hasn't the student done better than perfect?

Well, the student hasn't done something more perfect. But the student has taken her answers above and beyond the nature of a calculus exam. So, yes, while one cannot be more than perfect, one can go above nature. Perfection is not the same as maximality of value.

How did we come to justifiedly believe that there are three dimensions?

I think a really interesting philosophy of science project would be to ask how it is that we came to have the then-justified belief that there are three dimensions of space? (I don't know that that belief is still justified now given the serious possibility that String Theory is true.) Do any of my readers know anything on the intellectual history of the tridimensionality of space? I don't even know when it was first proposed. Euclid would be my guess.

A nominalist reduction

Suppose that there were only four possible properties: heat, cold, dryness and moistness. Then the Platonic-sounding sentences that trouble nominalists could have their Platonic commitments reduced away. For instance, van Inwagen set the challenge of how to get rid of the commitment to properties (or features) in:

1. Spiders and insects have a feature in common.

On our hypothesis of four properties, this is easy. We just replace the existential quantification by a disjunction over the four properties:

2. Spiders and insects are both hot, or spiders and insects are both cold, or spiders and insects are both dry, or spiders and insects are both moist.

And other sentences are handled similarly. Some, of course, turn into a mess. For instance,

3. All but one property are instantiated

becomes:

4. Something is hot and something is cold and something is dry but nothing is moist, or something is hot and something is cold and something is moist but nothing is dry, or something is hot and something is dry and something is moist but nothing is cold, or something is cold and something is dry and something is moist but nothing is hot.

Of course, this wouldn't satisfy Deep Platonists in the sense of this post, but that post gives reason not to be a Deep Platonist.

And of course there are more than four properties. But as long as there is a finite list of all the possible properties, the above solution works. But in fact the solution works even if the list is infinite, as long as (a) we can form infinite conjunctions (or infinite disjunctions—they are interdefinable by de Morgan) and (b) the list of properties does not vary between possible worlds. Fortunately in regard to (b), the default view among Platonists seems to be that properties are necessary beings.

Reverse Frankfurt cases

On standard Frankfurt cases, there is a counterfactual intervener who does nothing in the actual world, but who would prevent the action if one willed otherwise. I've been musing about reverse counterfactual interveners who do nothing in the actual world, but who would enable the action if one willed otherwise. For instance:

• Fred is sitting on the sofa watching The Good Guys. Unbeknownst to him, freak cosmic rays have just severed the nerve connections between his brain and his leg muscles. Fred knows the baby needs a change, but decides not to get up, and keeps on watching the show.

If that's the whole story, then:

1. Fred can't get up.
2. Fred is responsible for not trying to change the baby.
3. Fred is not responsible for his baby not being changed by him (since he can't change the baby).

But now add to the story:

• An alien monitoring Fred's thoughts would instantly reconnect the nerve connections as soon as Fred started trying to go change the baby.

The alien doesn't affect (2), of course. But does she affect (1) and (3)?

I have a hard time deciding whether Fred can get up with the alien in place. Consider:

• I don't try to run as fast as possible. But an alien is monitoring my thoughts, and were I to try to run as fast as possible, he would supercharge my muscles and the grippiness of my shoes and I'd run at Mach 3.

Can I run at Mach 3? There is something that I can do such that were I to do it, I would run at Mach 3. But maybe this doesn't make it be the case that I can run at Mach 3. Rather, maybe this just makes it be the case that I can do something that would make me able to run at Mach 3. After all, consider a very different case.

• I don't try to run as fast as possible. But an alien is monitoring my thoughts, and were I to try to run as fast as possible, he would make me able to speak Cantonese.

In this case, clearly I can't speak Cantonese, though there is something I can do such that were I to do it, I would become able to speak Cantonese. If this is like the Mach 3 case—and I am not completely sure of that—then in that case, I too can't run at Mach 3. And that suggests that even with the alien in place, Fred can't get up—though, again, I am not completely sure the Mach 3 case is like the Fred-and-alien case.

But perhaps the ability and responsibility don't line up. For I find it plausible that Fred is responsible for the baby not being changed by him in the case of the alien. After all, such double prevention things are not that unusual. To adapt Locke's example, you're at a party, and the host for security reasons locks the door but installs a doorman who will unlock the door who will open the door for anyone who wants to leave. It sure seems clear that if you stay at the party, you are responsible for that.

Maybe what happens is this. Assessment of outcome responsibility ("Is Fred responsible for the baby being unchanged by him?") tracks something like counterfactuals, while ability ("Can Fred get up?") tracks "internal features". The line between the two ways of tracking may not always be clear (I have some scepticism about how precisely defined counterfactuals are), but perhaps nothing of great moral significance rides on either one. For what matters morally for guilt and praiseworthiness is not what outcomes you are responsible for, but only what choices, what acts of will or failures to will, what tryings and failures to try, you are responsible for. Outcome responsibility does matter for the court system, especially but not only in civil cases, but that's mainly a matter of policy.

If that's right, then it does something interesting to some of the dialectics about alternate possibilities. For instance, Peter van Inwagen has argued that determinism and Frankfurt-style interveners would take away one's responsibility for certain outcomes. The compatibilist can embrace this conclusion. For the morally important question is about responsibility for one's will, not for outcomes. It could in principle be that we are responsible for no outcomes (if only because it could be that our acts of will have no outcomes), but we are responsible for our will.

But I don't know that this gets the compatibilist off the hook entirely. For something like an ability to try is important to assessing responsibility for a failure to try. And it is not clear that compatibilists have very good accounts of the ability to try.

Ordering subsets of the line and the Banach-Tarski paradox

Suppose we want to rank the subsets of the interval [0,1] by size. Consider this:

1. There is a total, transitive and reflexive ordering (i.e., a total preorder) ≤ of subsets of the interval [0,1] such that (a) if A⊆B, then A≤B and (b) if A and B are disjoint non-empty sets with A≤B, then A<A∪B,

where X<Y means that X≤Y but not conversely.

Condition (a) is intuitively correct for any notion of size comparison. Condition (b), then, is like a regularity condition in probability theory. Any regular probability is going to yield a comparison like in (1).

One can get such an ordering by applying Szpilrajn's Theorem to get a total ordering of the subsets of [0,1] that extends subset inclusion. Note, however, that Szpilrajn's Theorem uses a version of the Axiom of Choice. It's an interesting question whether one can have (1) without any Axiom of Choice.

Now, the Banach-Tarski Paradox—that a solid ball can be disassembled into a finite number of subsets that can be reassembled into two solid balls each of the same size as the original—also uses a version of the Axiom of Choice. It would be nice to have (1) while avoiding the Banach-Tarski Paradox. One might even think Bayesian epistemology requires that one be able to hold on to (1) while avoiding the Banach-Tarski Paradox, since the latter spells doom for rigid-motion invariant probabilities in a three-dimensional region. But without using any version of the Axiom of Choice one can prove:

Theorem. If (1) is true, then the Banach-Tarski Paradox holds.

This essentially follows from Note 1 in Pawlikowski's paper, together with the fact that the solid ball has the same cardinality as [0,1] (so an order on the subsets of [0,1] as in (1) transfers to an order on the subsets of the ball that satisfies (1)), and the following pleasant observation:

2. If ≤ is as in (1) and A₁,A₂,A₃,A₄ are non-empty sets, then if B=A₁∪A₂∪A₃∪A₄, then at most one of the A_i satisfies the condition B−A_i<A_i and at least one of the A_i satisfies the condition A_i<B−A_i.

Say that X~Y iff X≤Y and Y≤X. To prove (2), suppose first that A_i>B−A_i and j≠i. Then A_j⊆B−A_i. Moreover, A_i⊆B−A_j. Thus, B−A_j≥A_i>B−A_i≥A_j, and so we do not have A_j>B−A_j. Next, suppose A_i does not satisfy A_i≤B−A_i. Thus, by totality of ≤, we have A_i>B−A_i. But this can happen for at most one i. So there will be three distinct i such that A_i≤B−A_i. To obtain a contradiction, suppose all three inequalities are non-strict and suppose i and j are two of the three indices. Then A_i~B−A_i and A_j~B−A_j. Observe that A_j is a subset of B−A_i, and hence A_j≤A_i. By the same argument, A_i≤A_j, and so A_j~A_i. The same will go for the third index, so there are three indices i, j and k such that A_i~B−A_i~A_j~B−A_j~A_k~B−A_k. But B−A_k contains the union of A_i and A_j and so by (1) we have A_i<B−A_k, which is a contradiction.

Corollary 1. The claim that every partial order extends to a total order implies the Banach-Tarski Paradox.

And since it's well known that the Banach-Tarski Paradox cannot be proved in Zermelo-Fraenkel (ZF) set theory unless ZF is inconsistent:

Corollary 2. Claim (1) cannot be proved in ZF, unless ZF is inconsistent.

I have some thoughts on how one might perhaps be able to improve Corollary 1 to show the result under the weaker assumption that every set has a total order. (The existence of Lebesgue nonmeasurable sets can be proved from that.)

Philosophical implications: I think standard Bayesianism requires both (1) and the falsity of Banach-Tarski. So standard Bayesianism fails. Moreover, we get an argument for the possibility of incommensurability in decision theory. For if (1) is false, then there can be incommensurability (denying (1) requires affirming that there are some partially preordered sets that have no total preorder whose strict order relation extends the strict order relation of the original preorder), and if Banach-Tarski is true, there can be incommensurability (incommensurability in probabilities implies incommensurability in choices).

Introducing doppelgangers

Yesterday, I mentioned that one might reinterpret the quantifier symbols in a language so as to introduce doppelgangers: extra quasi-entities that just don't exist, but get to be talked about using standard quantifier inference rules. Here I want to give a bit more detail, and then offer a curious application to the philosophy of mind: an account of how a materialist could use a doppelganged reinterpretation of language to talk like a hard-core dualist. The application shows that it is philosophically crucial that we have a way to distinguish between real quantifiers and mere quasi-quantifiers (in the terminology of the previous post) if we want to distinguish between materialism and dualism.

Onward! Let L be a first-order language with identity. A model M for L will be a pair (D,P), where D is a non-empty set ("domain") and P is a set of subsets ("properties") of D. A doppel-interpretation of L is a pair (I,s) where I is a function from the names and predicates other than identity to D and P respectively and s is a function from names to the set {0,1} ("signature"). The signature function s tells us which name is attached to an ordinary object (0) and which to a doppelganger (1).

Now a substitution vector for a doppel-interpretation (I,s) will be a partial function v from the names and variables of L to D such that v(a)=I(a) whenever a is a name. A signature vector is a partial function f from the names and variables of L to D such that f(a)=s(a) whenever a is a name. If x is a variable and u is in D, then I will write v(x/u) for the substitution vector that agrees with v except that v(x/u)(x)=u. I.e., v(x/u) takes v and adds or changes the substitution of u for x. Likewise, if f is a signature vector, then s(x/n) agrees with s except that s(x/n)(x)=n for n in {0,1}.

We can now define the notion of a substitution and signature vector pair (v,f) doppel-satisfying a formula F in L under a doppel-interpretation (I,s). We begin with the normal Tarskian inductive stuff for truth-functional connectives:

• (v,f) doppel-satisfies F or G iff (v,f) doppel-satisfies F or (v,f) doppel-satisfies G
• (v,f) doppel-satisfies F and G iff (v,f) doppel-satisfies F and (v,f) doppel-satisfies G
• (v,f) doppel-satisfies ~F iff (v,f) doesn't doppel-satisfies F.

Now we need our quasi-quantifiers:

• (v,f) doppel-satisfies ∃xF iff for some u in D, (v(x/u),f(x/0)) doppel-satisfies F or for some u in D, (v(x/u),f(x/1)) doppel-satisfies F
• (v,f) doppel-satisfies ∀xF iff for every u in D, (v(x/u),f(x/0)) doppel-satisfies F and for every u in D, (v(x/u),f(x/1)) doppel-satisfies F.

Then we need atomic formulae:

• (v,f) doppel-satisfies h=k (where h and k are variables-or-names) iff v(h)=v(k) and f(h)=f(k)
• (v,f) doppel-satisfies Q(h₁,...,h_n) where Q is other than identity iff (I(h₁),...,I(h_n))∈I(Q).

And finally we can define doppel-truth: a sentence S of L is doppel-true provided that it is doppel-satisfied by every substitution and signature vector pair.

It is easy to see that if s_m is the usual sentence that asserts that there are m objects, and if D has n objects, then s_m is doppel-true if and only if m=2n. It is also easy to see that all the first order rules for quantifiers are valid for our quasi-quantifiers ∃ and ∀.

Now on to our fake dualism. We need a more complex doppelganging. Specifically, we need to divide our stock of predicates into the mental and non-mental predicates. Start with a materialist's first order language L that includes mental predicates (the materialist may think they are in some sense reducible). Add a predicate SoulOf(x,y) (we won't need to specify whether it's mental or not) which will count for us as neither mental nor non-mental. I will assume for simplicity that the mental predicates are all unary (e.g., "thinks that the sky is blue")—things get more complicated otherwise, but one can still produce the fake dualism. Now, instead of doppelganging all the objects, we only doppelgang the minded objects. Thus, our models will be triples (D,D_m,P) where D_m is a subset of D (the minded objects, in the intended interpretation). We say that a substitution and signature vector pair (v,f) is licit if and only if f(a)=1 implies v(a)∈D_m ("only members of D_m have doppelgangers"), and in all our definitions we only work with licit pairs. Moreover, our reinterpretations do not need to give any extension to the predicate SoulOf(x,y): that's handled in the semantics.

Finally, we modify doppel-satisfaction for quantifiers and predicates:

• (v,f) doppel-satisfies ∃xF iff for some u in D, (v(x/u),f(x/0)) doppel-satisfies F or for some u in D_m, (v(x/u),f(x/1)) doppel-satisfies F
• (v,f) doppel-satisfies ∀xF iff for every u in D, (v(x/u),f(x/0)) doppel-satisfies F and for every u in D_m, (v(x/u),f(x/1)) doppel-satisfies F.
• (v,f) doppel-satisfies h=k (where h and k are variables-or-names) iff v(h)=v(k) and f(h)=f(k)
• (v,f) doppel-satisfies Q(h₁,...,h_n) where Q is non-mental and other than identity iff (I(h₁),...,I(h_n))∈I(Q) and f(h₁)=...=f(h_n)=0
• (v,f) doppel-satisfies Q(h) where Q is mental iff I(h)∈I(Q) and f(h)=1
• (v,f) doppel-satisfies SoulOf(h,k) iff v(h)=v(k), f(h)=1 and f(k)=0.

And then we say we have doppel-truth of a sentence when every licit pair is satisfied.

The intended materialist doppel-interpretation (I,f) consists of the usual materialist interpretation I of the names and predicates other than Soul(x) and that gives to Soul(x) the extension of all the minded objects, sets D_m to be the set of minded objects, plus has a signature f such that f(n)=1 where n is a name of a minded object and otherwise f(n)=0.

Now let's speak an informal version of our doppelganged materialist language. Let M be any mental predicate and Q any non-mental one. Say that a soul is any x such that ∃y(SoulOf(x,y)). Suppose "Jill" is the name of a minded object. The following sentences will be doppel-true:

• Some objects have souls.
• Every object that has a soul is a non-soul.
• Only souls satisfy M.
• No soul satisfies Q.
• Jill is a soul.
• Jill does not satisfy Q.

The doppel-language is strongly dualist. Only the souls have mental properties predicated of them and only the non-souls have non-mental properties (or stand in non-mental relations). A materialist community could stipulate that henceforth their language bears this kind of doppel-interpretation. They could then talk like dualists. But they wouldn't be dualists.. Hence the doppelganged ∃ and ∀ aren't really quantifiers.

Assume materialism. If there are n material objects, including m minded objects, then in the doppelganged language it will be true to say something like "There are n+m objects." For every one of the minded objects has a doppelganger.

My favorite Aquinas quote

Hence we must say that the distinction and multitude of things come from the intention of the first agent, who is God. For He brought things into being in order that His goodness might be communicated to creatures, and be represented by them; and because His goodness could not be adequately represented by one creature alone, He produced many and diverse creatures, that what was wanting to one in the representation of the divine goodness might be supplied by another. For goodness, which in God is simple and uniform, in creatures is manifold and divided and hence the whole universe together participates the divine goodness more perfectly, and represents it better than any single creature whatever. (S.Th. I.47.1)

Quasi-quantifiers

In First Order Logic (FOL), there are three aspects to a quantifier:

• grammar: a quantifier attaches to a formula and generates a new formula binding one variable
• inference: we have the FOL universal and existential introduction and elimination rules
• semantics: the Tarskian definition of truth in a model treats quantifiers in a particular way with respect to a domain.

The first two aspects are normally lumped together under "syntactic aspects", but I think keeping them separate is important.

A quasi-quantifier, then, is something has the grammar and inferential structure of a quantifier, but may have different semantics. Every quantifier is also a quasi-quantifier. A quasi-quantifier that isn't a quantifier—i.e., that has aberrant semantics—will be a quantifier. Quasi-quantifiers can be of types, like "existential" or "universal", that correspond to those of quantifiers. One can have formal languages with existential and universal quasi-quantifiers. In fact, to an approximation English is a language with quasi-quantifiers: "there is" is a mere quasi-quantifier. I will argue for the possibility of mere quasi-quantifiers, connect the issue with fundamentality and then make my suggestion about English.

For any natural number n and quantifier E, let s_n(E) be the analogue of the FOL sentence using ∃ that asserts that there are n objects. For instance, s₂(E) is the sentence

• ExEy(x≠y&~Ez(z≠x&z≠y)).

A sufficient condition for E to be an existential mere quasi-quantifier is that E has the grammar and inferential rules of ∃ but is interpreted in such a way that s_n(E) is false in some model with a domain with n objects.

An uninteresting way to get an existential mere quasi-quantifier is by domain restriction. Restrict interpretations in such a way that names must all be in a subdomain of the model and quantifiers are restricted to the subdomain. A non-trivial quasi-quantifier is a mere quasi-quantifier that isn't just a restricted quantifier.

A sufficient condition for E to be an existential non-trivial quasi-quantifier is that E has the grammar and inferential rules of ∃ but is interpreted in such a way that s_n(E) is true in some model with a domain with fewer than n objects.

It isn't hard to generate languages with interpretations that make them have non-trivial quasi-quantifiers, though we will have to reinterpret the identity as well. For instance, it's not hard to generate a pair of existential and universal "doppelganging quantifiers"[note 1], that have the same inferential rules as the existential and universal quantifiers, but a sentence gets interpreted in a model as if each item in the model had a doppelganger, where a doppelganger of x stands in the same relations as x, except for identity (x=x but x's doppelganger isn't identical with x), and yet without adding any objects to the domain.[note 2]

Whether a quasi-quantifier is a quantifier depends on how that quasi-quantifier is treated in a Tarski-style definition of truth. Now, when we quasi-quantify also over non-fundamental objects, like holes and shadows, I think the Tarski-style definition of truth will give the truth conditions in terms of how the fundamental objects are (say, perforate or shadowing). This is going to be controversial, but, hey, this is only a blog post.

It follows immediately that when we quasi-quantify also over non-fundamental objects, we have a mere quasi-quantifier. Moreover, it's not going to be a restricted quantifier, so it's a non-trivial quasi-quantifier.

Now the English "there is" to an approximation is a quasi-quantifier. (It's not quite a quasi-quantifier, as the rules of inference for it will not quite match that of ∃ due to vagueness.) Moreover, it quasi-quantifies also over things like holes and defects and chairs, which are non-fundamental. Therefore, it is a mere quasi-quantifier. Nor is it just a restriction of a quantifier, so it is a non-trivial quasi-quantifier.

Once we see this, temptations to quantifier pluralism should be decreased. Of course, we have quasi-quantifier pluralism: There are quantifiers, there are doppelganging quasi-quantifiers, there are English quasi-quantifiers, there are mereological quasi-quantifiers, and so on. But only the first of these are quantifiers.

Now, in the formal examples, like of my doppelganging quantifiers, one can give a paraphrase of the quasi-quantifiers in terms of quantifiers: one just writes out the Tarski definition of truth for each sentence. But in natural language examples, the Tarski definition of truth is not going to be formally statable (at least not in any way tractable to us). And so there won't be a paraphrase of the quasi-quantifier sentences in quantified sentences. Quine won't like that. And what I said above about the Tarski definition when I characterized quasi-quantifiers won't be easy to say in the natural language case. There is much more work to be done here.

And of course just as there is no entity without identity, there is no quasi-quantifier without quasi-identity.